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In this paper, a Quadrature by Two Expansions (QB2X) numerical integrationtechnique is developed for the single and double layer potentials of theHelmholtz equation in two dimensions. The QB2X method uses both local complexTaylor expansions and plane wave type expansions to achieve a resultingrepresentation which is numerically accurate for all target points (interior,exterior, or exactly on the boundary) inside a leaf box in the fast multipolemethod (FMM) hierarchical tree structure. Compared to the original Quadratureby Expansion (QBX) method, the QB2X method explicitly includes the nonlinearityfrom the boundary geometry in the plane wave expansions, thereby providing forhigher order representations of both the boundary geometry and densityfunctions in the integrand, with its convergence following standard FMM erroranalysis. Numerical results are presented to demonstrate the performance of theQB2X method for Helmholtz layer potentials and its comparison with the originalQBX method for both flat and curved boundaries with various densities. The QB2Xmethod overcomes the challenges of the original QBX method, and is bettersuited for efficient solutions of the Helmholtz equation with complexgeometries.

Quadrature by Two Expansions for Evaluating Helmholtz Layer Potentials